The Conversation Inbox displays messages specific to your user profile. This means that you will not be able to see conversations relating to messages send from other users - only your own.
You can access the Conversation inbox by clicking Conversations in the navigation pane:
You’ll be taken to a page where you can view the SMS conversations you’ve had with recipients, showing the sent and received messages in a clear, concise format.
On the left you’ll be able to see a snippet of your conversations, and when you click it, on the right you’ll be able to see the full conversation. This reading pane allows you to easily have a two-way conversation with the recipient. You’ll be able to use this page to see if your message has been delivered or not.
You have the ability to archive messages that you no longer need to see in the list by clicking the Archive button on the top right:
You can view archived messages by clicking on the Archived tab over on the left above the contact search bar.
You can also filter your inbox for read and/or unread messages using these tabs: